
Analytical expressions of the nucleon specific energy \(E_0(\rho )\) , pressure \(P_0(\rho )\) , incompressibility coefficient \(K_0(\rho )\)  and skewness coefficient \(J_0(\rho )\)  of SNM are derived in a general \(d\) D space. The corresponding expressions for the quadratic symmetry energy \(E_{\mathrm {sym}}(\rho )\) , its slope parameter \(L(\rho )\)  and curvature coefficient \(K_{\mathrm {sym}}(\rho )\)  as well as the fourth-order symmetry energy \(E_{\mathrm {sym,4}}(\rho )\)  are also given.

The general features of the EOS of ANM in 2D are analyzed in some details. In particular, we found that the EOS of ANM (including the symmetry energy and the EOS of SNM) could be obtained from an effective expansion based on \(\sqrt{\rho }\) . Moreover, there are no kinetic terms appearing in the general expressions for \(K_0(\rho ), J_0(\rho )\)  and \(E_{\mathrm {sym,4}}(\rho )\) . Based on this finding, we have shown that the quartic term \(E_{\mathrm {sym,4}}(\rho )\)  is generally far smaller than the quadratic symmetry energy \(E_{\mathrm {sym}}(\rho )\)  in the sense that \(E_{\mathrm {sym,4}}(\rho _0)/E_{\mathrm {sym}}(\rho _0)\lesssim 1\%\) , validating the conventional parabolic approximation for the EOS of ANM in 2D.
Furthermore, the fourth-order symmetry energy in other dimensions (from 1D to higher dimensions) is also shown to be small due to the specific structure of \(E_{\mathrm {sym,4}}(\rho )\)  where the dimension \(d\)  plays an essential role (see the expression ()). Thus, from the viewpoint in a \(d\) D space in which the nucleons live, the effectiveness of the parabolic approximation of its EOS is consequently natural and the conventional 3D case is not unique.

The EOSs in \(d\) D spaces are derived from the \(\epsilon \) -expansion (using an assumed small perturbative dimension \(\epsilon =d-d_{\mathrm {f}}\)  with \(d_{\mathrm {f}}\)  being a reference dimension) based on the EOS in 3D.
In particular, we investigated the kinetic EOS both for the FFG model (section  with \(d_{\mathrm {f}}=3\)  to linear order of \(\epsilon \) ) and for the HMT model (section  with general \(d_{\mathrm {f}}\) ). We found that starting from the conventional 3D kinetic EOS, one can reasonably approximate the corresponding kinetic EOS of SNM and the kinetic symmetry energy in 2D, see (REF ) and (REF ). In addition, the \(\epsilon \) -expansion based on \(d_{\mathrm {f}}\approx 1\sim 3\)  even with the SRC-induced HMT included is effective, see results of Fig. REF .
Moreover, the tendency of the EOS in a perturbed dimension is analyzed using the relevant analytical expressions.
Specifically, we found that the nucleon specific energy and pressure in SNM will be reduced (enhanced) if the perturbative dimension \(\epsilon \)  is negative (positive) compared with the 3D case, e.g., the EOS of SNM in 2D will be correspondingly softened compared with the 3D EOS of SNM, see the formula (REF ). The correction to the symmetry energy from the \(\epsilon ^2\)  contribution does not qualitatively change the prediction from the linear approximation, as shown in section .
In a nutshell, the 3D EOS of ANM in fact encapsulates the very relevant/useful information on the EOS in \(d_{\mathrm {f}}\) D with \(d_{\mathrm {f}}\)  being near 3, and the \(\epsilon \) -expansion provides a useful technique to explore the low-\(d\)  EOS of ANM from its 3D counterpart.

A toy model for the single-nucleon potential is constructed by enforcing the corresponding EOS of ANM to fulfill all the available empirical constraints, including the one on the pressure in SNM from studying nuclear collective flow data in relativistic heavy-ion collision [1]}, the constraint on the single-nucleon potential in SNM from optical model analyses of nucleon-nucleus scattering data [2]}, the scalar nucleon effective mass around the saturation density [3]}, the causality constraint at densities \(\lesssim 6\rho _0\) , the isobaric analog state constraint on the symmetry energy near the saturation density [4]} as well as the constraint on the EOS of PNM from chiral effective field theories [5]}. The constructed model (which owns certain momentum dependence) is then applied to study the EOS of ANM in \(d\) D spaces. Specifically, we found that the EOS of SNM is enhanced (reduced) if \(d\)  is upwardly (downward) perturbed with respect to the 3D case, verifying the general conclusion found in section .
This means that the many-nucleon system tends to be more bounded in spaces with reduced dimensions, while it may become completely unbounded in spaces with \(\gtrsim 4\) . The latter phenomenon in fact reflects some deep principles from the viewpoint of the high-dimensional geometry regarding the \(d\) -dimensional sphere.
In addition, the symmetry energy in 2D is found to be reduced compared to its correspondence in 3D.

The kinetic EOS of ANM in the presence of SRC-induced HMT in the single-nucleon momentum distribution function in \(d\) D is studied. Besides the effectiveness of the \(\epsilon \) -expansion of the kinetic EOS of SNM, we also found that the first two moments of the nucleon momentum distribution in lower dimensions become smaller in both the FFG and HMT models, while the relative nucleon momentum fluctuation significantly increases as \(d\)  decreases below 2, see Fig. REF .
More interestingly, the prediction on the relative momentum fluctuation \(\Upsilon _k\)  from the HMT and the FFG model becomes closer as \(d\)  decreases, indicating that the many-nucleon system in low dimensions behaves like a free system in terms of their kinetic EOSs. The same phenomenon is also found through the calculations on the isospin-expansion of the kinetic EOS of ANM, i.e., the predictions on them from the FFG and the HMT model become very similar in low dimensions (e.g., in 1D or 2D), as shown in the left panel of Fig. REF . The nearly free property of the kinetic EOS of ANM in low dimensions may provide a useful tool to explore the EOS in 3D from the corresponding counterpart in 2D or 1D, or vice versa.
Furthermore, the predictions on the kinetic EOS of PNM by combining the \(E_0^{\mathrm {kin}}(\rho )\) , \(E_{\mathrm {sym}}^{\mathrm {kin}}(\rho )\) , \(E_{\mathrm {sym,4}}^{\mathrm {kin}}(\rho )\)  and \(E_{\mathrm {sym,6}}^{\mathrm {kin}}(\rho )\)  from the HMT and the FFG models are very close for \(d\lesssim 3\) , verifying the conventional picture in 3D that there exist very few high-momentum neutrons in PNM.
However, as the dimension \(d\)  increases beyond 3, the above predictions from the HMT and the FFG models tend to deviate, see the right panel of Fig. REF , indicating that the EOS of PNM in high dimensions may behave very differently from the one in \(d\lesssim 3\) .

